(This article was originally published at Error Statistics Philosophy » Statistics, and syndicated at StatsBlogs.)

The recent post by Normal Deviate, and my comments on it, remind me of why/how I got back into the Bayesian-frequentist debates in 2006, as described in my first “deconstruction” (and “U-Phil”) on this blog (Dec 11, 2012):

Some time in 2006 (shortly after my ERROR06 conference), the trickle of irony and sometime flood of family feuds issuing from Bayesian forums drew me back into the Bayesian-frequentist debates.^{1 2} Suddenly sparks were flying, mostly kept shrouded within Bayesian walls, but nothing can long be kept secret even there. Spontaneous combustion is looming. The true-blue subjectivists were accusing the increasingly popular “objective” and “reference” Bayesians of practicing in bad faith; the new O-Bayesians (and frequentist-Bayesian unificationists) were taking pains to show they were not subjective; and some were calling the new Bayesian kids on the block “pseudo Bayesian.” Then there were the Bayesians somewhere in the middle (or perhaps out in left field) who, though they still use the Bayesian umbrella, were flatly denying the very idea that Bayesian updating fits anything they actually do in statistics.^{3} Obeisance to Bayesian reasoning remained, but on some kind of a priori philosophical grounds. Doesn’t the methodology used in practice really need a philosophy of its own? I say it does, and I want to provide this.

The result of my own interest here gave rise to a Kent-Virginia Tech workshop (in Kent)^{4} followed by the 2010 conference at the LSE, from which grew the special volume (see Mayo 2010, RMM volume, for examples and references)….

Especially surprising to me, leaders of default Bayesianism, arguably the most predominant current form, began claiming that “violation of principles such as the likelihood principle is the price that has to be paid for objectivity” (Berger 2006, 394). As such, the default Bayesian may welcome with relief my critique of Birnbaum’s famous LP argument. (See my December 6 and 7 posts, and our current U-Phil) Even though “objectivity” is used very differently, there is still this odd sort of agreement in phrases uttered. While for us the violation is fully in order and is picked up on through the sampling distribution; for Bayesians it is anything but expected, and is picked up through model-dependent changes of priors (introducing strict incoherence).

It is noteworthy that default Bayesians don’t agree with each other even with respect to standard applications, as they readily admit. For instance, Bernardo, but not Berger, rejects the spiked prior that leads to pronounced conflicts between frequentist p-values and posteriors. While reonuncing the spikes makes the numbers agree (with frequentists), there is no evidence that the result is either an objective or rational degree of belief (as he intends) or an objective assessment of well-testedness (as our error statistician achieves). Bernardo wants to match the frequentist in the optional stopping case, but I take it Jim still adheres to the position of Berger and Wolpert 1988 on the SRP.

OK, so here’s an especially intriguing remark by Jim Berger that I think bears upon the current mindset. (Jim is aware of my efforts, it will come as no surprise that I’m sharing my meandering here.)

Too often I see people pretending to be subjectivists, and then using “weakly informative” priors that the objective Bayesian community knows are terrible and will give ridiculous answers; subjectivism is then being used as a shield to hide ignorance. . . . In my own more provocative moments, I claim that the only true subjectivists are the objective Bayesians, because they refuse to use subjectivism as a shield against criticism of sloppy pseudo-Bayesian practice. (Berger 2006, 463)

How might we deconstruct this fantastic remark of Berger’s?^{5} (Granted, this arises in his rejoinder to others, but this only heightens my interest in analyzing it.)

Here, “objective Bayesians” are understood as using (some scheme) of default or conventionally derived priors. One aspect of his remark is fairly clear: pseudo-Bayesian practice allows “terrible” priors to be used, and it would be better for them to appeal to conventional “default” priors that at least will not be so terrible (but in what respect?). It is the claim he makes in his “more provocative moments” that really invites deconstruction. Why would using the recommended conventional priors make them more like “true subjectivists”? I can think of several reasons—but none is really satisfactory, and all are (interestingly) perplexing. I am reminded of Sartre’s remarks in Being and Nothingness on bad faith and irony:

In irony a man annihilates what he posits within one and the same act; he leads us to believe in order not to be believed; he affirms to deny and denies to affirm; he creates a positive object but it has no being other than its nothingness.

So true! (Of course I am being ironic!) Back to teasing out what’s behind Berger’s remarks.

Now, it would seem that if she did use priors that correctly reflected her beliefs (call these priors “really informed by subjective opinions”(riso?), and that satisfied the Bayesian formal coherency requirements, then that would be defensible for a subjective Bayesian. But Berger notices that, in actuality, many Bayesians (the pseudo-Bayesians) do not use riso priors. Rather, they use various priors (the origin of which they’re unsure of) as if these really reflected their subjective judgments. In doing so, she (thinks that she) doesn’t have to justify them—she claims that they reflect subjective judgments (and who can argue with them?).

According to Berger here, the Bayesian community (except for the pseudo-Bayesians?) knows that they’re terrible, according to a shared criterion (is it non-Bayesian? Frequentist?). But I wonder: if, as far as the agent knows, these priors really do reflect the person’s beliefs, then would they still be terrible? It seems not. Or, if they still would be terrible, doesn’t that suggest a distinct criterion other than using “really informed” (as far as the agent knows) opinions or beliefs?…

RMM 2011 refers to the special issue of the on-line journal, Rationality, Markets and Morals housing papers growing out of the LSE conference of June 2010: Statistical Science and Philosophy of Science: Where Do (Should) They Meet in 2011 and Beyond? http://www.rmm-journal.de/htdocs/st01.html

To read follow-up blogposts to this call for U-Phils:

Contributed deconstructions of J. Berger:http://errorstatistics.com/2011/12/26/contributed-deconstructions-irony-bad-faith-3/

J. Berger on J. Berger:http://errorstatistics.com/2011/12/29/jim-berger-on-jim-berger/

[1] It was David Cox who first alerted me. Then there was Dongchu Sun, a statistician who was at Virginia Tech for a few years.

[2] Yes, I’d given up on them, and was happy to spend all my remaining exiled days on philosophy of experiment.

[3] I’m not here including things like “Bayes nets,” which use conditional probability (as do we all) but are not really Bayesian.

[4] J. Williamson, J. Corfield and others at Kent co-hosted the first.

[5] As noted, Jim Berger is aware that I’m discussing this on my blog. I hope he will comment!

Berger, J. (2006),“The Case for Objective Bayesian Analysis”, and “Rejoinder”, *Bayesian Analysis* 1(3), 385–402; 457-464.

Mayo, D. (2011), “Statistical Science and Philosophy of Science: Where Do/Should They Meet in 2011 (and Beyond)?” RMM Vol. 2, 2011, 79–102

Sartre, J.P *Being and Nothingness: an essay in phenomenological ontology* (1943, Gallimard); English 1956, Philosophical Library Inc.

Senn, S. (2011) You may believe you are a Bayesian but you are probably wrong. Rationality, Markets and Morals RMM Vol. 2, 2011, 48–66

Filed under: Likelihood Principle, objective Bayesians, Statistics Tagged: default priors, frequentist-Bayesian unifications, Jim Berger, objective Bayesians, pseudo-Bayesians

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